2024
Functional Analysis
Name: Functional Analysis
Code: MAT00904L
6 ECTS
Duration: 15 weeks/156 hours
Scientific Area:
Mathematics
Teaching languages: Portuguese
Languages of tutoring support: Portuguese
Presentation
In this optional discipline taught in the 5th semester, it is intended that the student has a learning of efficient abstract methods of analysis and functional equations and its implementation in significant examples of application to several specific areas of Analysis.
Sustainable Development Goals
Learning Goals
Objectives:
1. Basic formation in Functional Analysis.
2. Learning of the efficient abstract methods of functional analysis & functional equations, following the tradition initiated by Banach, and its concretization into significant examples of application to diverse specific areas of Analysis, such as differential & integral equations, numerical analysis,
Competencies:
- Develop abstract thought as a means of solving, with both greater generality & simplicity, specific problems of Analysis.
- Abstraction skills, creative intuition, model construction and spirit of criticism.
- Skills for explaining the obtained results, both orally and written.
1. Basic formation in Functional Analysis.
2. Learning of the efficient abstract methods of functional analysis & functional equations, following the tradition initiated by Banach, and its concretization into significant examples of application to diverse specific areas of Analysis, such as differential & integral equations, numerical analysis,
Competencies:
- Develop abstract thought as a means of solving, with both greater generality & simplicity, specific problems of Analysis.
- Abstraction skills, creative intuition, model construction and spirit of criticism.
- Skills for explaining the obtained results, both orally and written.
Contents
Finite and infinite dimensional Banach spaces.
Hahn-Banach theorem.
Strong and weak convergence.
Hilbert spaces.
Bounded linear operators. Compact operators. Fixed point theory.
Integral equations. Fredholm alternative.
Resolvent and spectrum.
Hahn-Banach theorem.
Strong and weak convergence.
Hilbert spaces.
Bounded linear operators. Compact operators. Fixed point theory.
Integral equations. Fredholm alternative.
Resolvent and spectrum.
Teaching Methods
Teaching methodologies:
- Structured exposition, examples with emphasis on applications and on solving exercises.
- To stimulate students initiative, so that classes become essentially centered on students activities, guided by their teacher; instead of on teachers activities, copied by students. Particularly in what concerns submission of questions and / or suggestions of application and / or description of contents, the solving of exercises, participation in discussions, etc
The evaluation may be either continuous, done through between two and six partial tests and quizzes, done
preferably during the classes, weighting 100% of the classification, the number of which is to be defined by
the professor who is responsible for the course unit in each academic year, or by a final exam.
- Structured exposition, examples with emphasis on applications and on solving exercises.
- To stimulate students initiative, so that classes become essentially centered on students activities, guided by their teacher; instead of on teachers activities, copied by students. Particularly in what concerns submission of questions and / or suggestions of application and / or description of contents, the solving of exercises, participation in discussions, etc
The evaluation may be either continuous, done through between two and six partial tests and quizzes, done
preferably during the classes, weighting 100% of the classification, the number of which is to be defined by
the professor who is responsible for the course unit in each academic year, or by a final exam.
Certificações
